Area and units of area
Area is the amount of flat space inside a shape. We count how many unit squares fit inside. A square with sides of 1 cm has area 1 cm² (one square centimetre).
Units of area are squares of units of length, so conversions use squares too:
- 1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm²
- 1 km = 1000 m, so 1 km² = 1 000 000 m²
- 1 hectare (ha) = 10 000 m², often used for fields
Always make sure all lengths are in the same unit before you multiply.
Rectangle and parallelogram
A rectangle with base b and height h holds b rows of h squares, so area = b × h.
A parallelogram has slanted sides. Cut a right triangle from one slanted end and slide it to the other end. The pieces fit together to make a rectangle with the same base and the same height. So area = base × height. The height is the straight (perpendicular) distance between the two parallel sides, not the slanted side.
Triangle
Take an exact copy of a triangle and turn it by 180° about the midpoint of one side. The two triangles join into a parallelogram with the same base and height. The triangle is half of it:
Area = ½ × base × height
Any side can be the base. The height is then the perpendicular distance from the opposite corner to that base.
Rhombus
A rhombus has four equal sides. Its two diagonals d₁ and d₂ cross at 90°. Draw a rectangle d₁ by d₂ around it so that the rhombus corners touch the middle of each rectangle side. The four corner triangles outside the rhombus fit exactly into the rhombus. So the rhombus is half of the rectangle:
Area = ½ × d₁ × d₂
This works for any shape whose diagonals cross at right angles, such as a kite.
Trapezium
Take a copy of the trapezium and turn it by 180° about the midpoint of one leg. The two trapeziums join into a parallelogram. Its base is (a + b) and its height is h. The trapezium is half of it:
Area = ½ × (a + b) × h
The same result is midline × height, because the midline is (a + b) ÷ 2.
Area of any polygon and proving formulas
For an odd polygon, split it into rectangles, triangles and trapeziums, find each area, and add them. To prove a formula, use cut-and-move or doubling as you saw above: show that the new shape is made of the same pieces, so the area cannot change.
Try it
Draw a parallelogram on graph paper, cut it out, then cut off a triangle from one side and slide it to the other side. You get a rectangle. Count the squares and compare with base × height. Now draw a triangle, cut out two copies, and join them into a parallelogram.
Key formulas and definitions
- Rectangle: A = b × h
- Parallelogram: A = base × height
- Triangle: A = ½ × base × height
- Rhombus: A = ½ × d₁ × d₂
- Trapezium: A = ½ × (a + b) × h
- 1 m² = 10 000 cm²; 1 ha = 10 000 m²; 1 km² = 1 000 000 m²
Worked examples
1. A field is 40 m long and 25 m wide. Find its area in m² and in hectares.
A = 40 × 25 = 1000 m². Since 1 ha = 10 000 m², it is 0.1 ha.
2. A parallelogram has base 9 cm and height 6 cm. Find its area.
A = base × height = 9 × 6 = 54 cm².
3. A triangle has base 10 cm and height 7 cm. Find its area.
A = ½ × 10 × 7 = 35 cm².
4. The diagonals of a rhombus are 8 cm and 6 cm. Find its area.
A = ½ × 8 × 6 = 24 cm².
5. A trapezium has parallel sides 7 cm and 5 cm and height 4 cm. Find its area.
A = ½ × (7 + 5) × 4 = ½ × 12 × 4 = 24 cm².
6. Convert 2.5 m² to cm². Then find the height of a trapezium of area 60 cm² with parallel sides 8 cm and 12 cm.
2.5 m² = 2.5 × 10 000 = 25 000 cm². For the trapezium: 60 = ½ × (8 + 12) × h = 10 × h, so h = 6 cm.
Common mistakes
- Using the slanted side instead of the height in a parallelogram or triangle. The height is at 90° to the base.
- Forgetting the ½ in the triangle, rhombus and trapezium formulas.
- Converting m² to cm² by multiplying by 100. The factor is 100 × 100 = 10 000.
- Multiplying lengths in different units, for example metres by centimetres, without converting first.